I'm going to attempt a rather long exposition of what's going on here because there are several issues involved.
First, it's important to note that this early section (§1) involves a deconstruction1 of traditional ontology. Hence, his point in this passage is that in the terms of traditional ontology, 'Being' cannot be conceived of as an entity. The quotation marks here are important, because they indicate that what is under discussion is not Being as such, but the concept of 'Being' as the supreme universality.
It follows from this "supreme universality" that 'Being' is not an entity in a rather straight-forward way. The traditional ontology Heidegger references requires that an entity has a definition, that is a genus and a specific differentiation.2 For example, the much abused Scholastic dictum that "Socrates is a man" involves both the genus (man) and the specific differentiation (Socrates), and hence is taken as a "full" definition of Socrates. 'Being', however, has no genus nor no specific differentiation since it is predicated of all things; likewise, 'Being' is not itself a genus nor is it itself a specific difference which defines the individual thing; hence "being adds nothing else to nature."3 Since 'Being' has no definition nor is involved in the definition of anything, 'Being' cannot be an entity.
This, I think, is sufficient for detailing how in terms of traditional ontology 'Being' cannot be an entity. But still the question you raise is an important one and, as @virmaior pointed out in the comments, Heidegger takes the idea that Being is not an entity to have an relevance beyond just the terms of traditional ontology.4 So, if we aren't thinking within the bounds of an Aristotelian notion of entity which requires both a genus and a specific differentiation, why can't Being be taken to be an entity? Or, as you wrote, "I can talk about it, study it, use it in various ways: It is and is not various things."
On my reading, Heidegger's initial answer to this question comes in the next numbered passage:5
It is held that 'Being' is of all concepts the one that is self-evident. Whenever one cognizes anything or makes an assertion, whenever one comports oneself towards entities, even towards oneself, some use is made of 'Being'; and this expression is held to be intelligible 'without further ado', just as everyone understands 'The sky is blue', 'I am merry', and the like. But here we have an average kind of intelligibility, which merely demonstrates that this is unintelligible. It makes manifest that in any way of comporting oneself towards entities as entities... there lies a priori an enigma. The very fact that we already live in an understanding of Being and that the meaning of Being is still veiled in darkness proves that it is necessary in principle to raise this question again (M&R p.23 / H4).
Here the problem of raising the question of the meaning of Being is put in terms outside the terminology of traditional ontology. Instead, Heidegger is simply referring to the everyday use of Being. The problem arises when we try to give an account of things. When we say "The sky is blue", everyone understands what this means. The copula "is" seems to add nothing but a grammatical niceity; one could very easily imagine constructing a sentence such as "the sky blue" or "blue the sky" and have it be perfectly understandable according to rules of a certain grammar.6 The problem that Heidegger sees is that taking Being as this empty signifier occludes the meaning of Being from our understanding; rather than 'simply understanding' Being, what such formulations show is that we do not grasp Being at all or rather, "we already live in an understanding of Being" without being able to articulate this understanding.
This "already liv[ing] in an understanding of Being" applies to how we come to think of Being itself. We can "study" Being and "use" Being "in various ways", but none of these really grasp Being itself. Rather what we study as Being is always simply a certain pre-conception7 of 'Being', a construct of our understanding, which, takes 'Being' as an empty, "obvious" signifier.
Given that the above is very abstract, let me try to make things more concrete. If I were to say "Being is..." or "Being is not...", for example, "Being is not an entity" then I face a kind of a priori problem because I can only make such an assertion about Being by already having a conception of Being, that is, by already assuming I know what this 'is' or 'is not' means. Hence, in these broader terms, I can only talk about how 'Being' 'is' or 'is not', i.e., how a certain conception of 'Being' operates in an already-given conception of 'Being', but not about Being itself.8
This a priori aporia seems to leave us with no hope of getting any closer to Being, but Heidegger thinks he has a way out: there is a primordial, pre-conceptual experience of Being that, when clarified can give us a non-conceptual meaning of Being. The task of Being and Time is to destroy our conceptions of Being in order to make this experience of Being come to light.
The question of space in Heidegger is a fairly complex one. In Being and Time Heidegger devotes §§22-24 to space and spatiality. This has been explored in numerous places (especially concerning architecture, topology and geography), but it's particularly worth mentioning the work done by Jeff Malpas and Stuart Elden. Suffice to say that for Heidegger, space is not an entity in much the same way that Being is not an entity. Most importantly, to take space as an entity is to operate with a conception of space, but this conception is ultimately grounded in the experience of spatiality, which, for Heidegger is what must first be clarified. This is, in effect, the general structure of Heidegger's phenomenology: everything that is has its Being grounded in a primordial structure of experience that becomes occluded by our conception of it.
The locus classicus for the discussion of entities (i.e., our primordial experience of entities) in Being and Time is §§15-18, which includes the famous discussion of tools and the difference between 'ready-to-hand' (Zuhandenheit) and 'present-at-hand' (Vorhandenheit). On this point, Graham Harman's Tool-Being could be a useful guide.
Notes
1: "Deconstruction" as used by Derrida derives from the use of "destruction" by Heidegger, in the sense outlined in §6 of Being and Time (p. 44-49 in Maquarrie & Robinson's translation, p. 22-27 in the German). "Deconstruction" in this sense is an attempt to call into question our pre-existing assumptions about a concept in order to open it up for new possibilities of understanding.
2: This is the upshot of the first Latin gloss Heidegger cites in this passage: definitio fit per genus proximum et differentiam specificam.
3: My translation of enti non additur aliqua natura. See Aristotle's argument in Metaphysics 3, especially 998b, where Aristotle argues that being is not a genus. For a use of these concepts in a Scholastic context, see, e.g., Aquinas' De Veritate q. 1 a. 1.
4: In Dreyfus' passage-by-passage commentary on Being and Time, Being-in-the-World, he notes that "[Heidegger] begins by noting three ways in which the nature of being has traditionally been miscontrued, ways that nonetheless contain hints of what he considers the real issues." (p. 11) While I have some issues with some of Dreyfus' interpretations, Being-in-the-World is a valuable and extremely clear commentary on Heidegger.
5: Being and Time is constructed in a rather difficult way where the initial hints of something to be developed are given first and the development of the idea occurs substantially later in the text. For example, this bit that I've called an "initial answer" is elucidated in §9, §16, §27, §31 etc., essentially spanning the entireity of the text.
6: This is, in fact, how ancient Greek and, in some cases, Latin actually functions; the copular verb is often omitted completely. One might also consider the various language-games Wittgenstein offers in Philosophical Investigations.
7: It would be slightly accurate to say "fore-conception" (Vorgriff) or "fore-having" (Vorhabe), two terms which become very important later in the text of Being and Time. But I'm trying to eschew Heideggerian jargon here.
8: If you like, I think it's fair to say that this conception of 'Being' (any conception whatsoever) is itself an entity. It can be defined, given bounds, discussed, etc. The point is that this conception of 'Being' still is in a certain way and both the way in which it is and what it means for it to be remain obscure; even more so since this conception stands in for what is trying to be explained.
