I realize that Seamus has already mentioned the classic criterion proposed by Quine, but I think Seamus may have slightly misrepresented it; so I thought I might add a few more details. It is true that Quine famously states "to be is to be the value of a variable" in is seminal paper "On What There Is." But I think it is slightly inaccurate to say that Quine thought that whatever exists is whatever is referenced/quantified in our best scientific theories, that the set of existing things just is the set of all that is quantified in our scientific theories. Here's what Quine says:
Now how are we to adjudicate among rival ontologies? Certainly the answer is not provided by the semantical formula "To be is to be the value of a variable"; this formula serves rather, conversely, in testing the conformity of a given remark or doctrine to a prior ontological standard. We look to bound variables in connection with ontology not in order to know what there is, but in order to know what a given remark or doctrine, ours or someone else's, says there is; and this much is quite properly a problem involving language. But what there is is another question.
In other words, Quine is saying that the criterion "to be is to be the value of a variable" is more or less a way of determining what are language commits us to ontologically, specifically, the referential/singular terms and quantified variables employed in language. So there may be existing things that may go 'undetected' by this criterion precisely because they haven't been referenced in our best scientific theories. For Quine, if there was no way of paraphrasing a instance of reference or quantification in a way that avoids that reference or quantification, then it has to exist. In fact, this is springboard for the Quine-Putnam Indispensability argument for Mathematical Platonism. Roughly, the argument is that, since our best physical theories contain existentially quantified propositions whose domain of quantification consists only of abstract objects, these abstract objects must exist.
I think Seamus is correct when he says that this is the view accepted by most philosophers, but that isn't to say it has gone entirely uncontested. As with any philosophical thesis, the Quinean criterion for ontological commitment has been challenged. E.g., William Lane Craig in his relatively new book "God and Abstract Objects" launches an attack on the Indispensability argument precisely by disputing the Quinean criterion for ontological commitment. I think William Alston, Jodi Azzouni, and Penelope Maddy are others who take issue with it too.
Now, let me take a stab at (partially) answering your question. Your criterion seems to precludes the existence of abstract objects. In the comments on Dheeraj Verma, user193319 brings up this very point. However, Dheeraj Verma seems to misunderstand what abstract objects are. He states that abstract objects exist through the mind. First of all, it isn't entirely clear what "existing through" consists in; this seems to mean they have existence by virtue of us thinking about them, which is a very mistaken view of abstract objects. Abstracts objects, if they exist, do so necessarily and independently of whether they are conceived of by contingent human minds. Moreover, we don't know abstract objects exist because we interact with them by thinking about them; this is impossible because abstract objects don't stand in causal relations. Rather, we know they exist because we posit their existence because they fulfill some role (e.g., the role of explaining why mathematical truths are true and why they are necessarily true). Finally, note that the Indispensability argument, if successful, shows that your 'sensory' criterion doesn't work, since sensory affection is ultimately what justifies science, yet this in turn shows that abstract objects exist, which are causally inert and therefore cannot interact with our senses.
In this edit I will try to answer the question Asmani posed in his comment. It gets tricky, however, since we begin to skirt issues in philosophy of mathematics and physics. Consider Newton's (differential) equation F=ma. You might imagine the following said in a physics course: "Such-and-such a particle is following this trajectory in space, and there exists a solution to Newton's equation describing its trajectory." These seem like to true statements. We might symbolize the last statement using logic in the following way: letting S be the set of all solutions
and "s is a solution describing the particle's trajectory", the last sentence reads Now, the basic idea behind this ontological principle is that the existential quantifier ranges over S, and S only consists of abstract object (namely, solutions to the differential equation). Since the statement is true, there has to be a solution (i.e., value of the bound variable) which makes it true and therefore there exists an abstract object.
I confess this is a somewhat artificial/contrived example, but I think it captures the gist of what "to be is to be the value of a variable" means and how this principle of ontological commitment is to be used; moreover, it shows how the Indispensability argument works. Indeed, I feel compelled to add that Quine himself would deny that this is a genuine example. Though Quine was a staunch Naturalist, he saw the need of abstract objects because of the success and truth of science; but he didn't want an 'extravagant' or 'inflated' ontology. Because he was able to reduce mathematics to set theory, he could see such things differential equations and their solutions as complicated logical constructions of set theory.
I'm afraid I can't go into all details on the process whereby higher mathematics are reducible to set theory. But, e.g., the natural numbers can be reduced to sets in the following way: 0 is taken to be identical to , the empty set; 1 identical to ; 2 identical to ; etc. From the naturals you construct the rest of the integer, rationals from the integers, etc. So in Quine's ontology, there aren't numbers but pure sets, since numbers are in a sense logical constructions.
Sorry for the extraordinarily long post--I have no one else to discuss philosophy with, so sometimes I go overboard; and sorry if this seems like a complicated example, but when I first read your comment, I was thinking you wanted an example in the context of the Indispensability argument.
In answer to Asmani's request about a physical object. Well, it's somewhat tricky. I think it depends upon context to some degree. Typically, using the word "this" connotes direct acquaintance with the referent of the demonstrative. If you are in the presence of 'this', then pointing to it and uttering "This chair exists" I think should be interpreted as meaning "There is a chair that I am pointing at", in which case there is a bound variable and 'chair' is substituted in for this variable (i.e., 'chair' is in the domain of the quantifier). if the sentence is true, then the chair must exist since the truth of the sentence commits us to the existence of it. Unless, however, there is some way of rephrasing the sentence that gets rid of this commitment, e.g., "There is matter arranged chairwise that I am pointing at." (such a paraphrase might be provided by a mereological nihilist, one who thinks there aren't literally such thing as tables, chairs, etc. but only physical particles or some other such simples arranged in various ways).