What kind of restrictions are you looking for, or are you thinking very broadly, fishing for options?
I think the best omnidirectional, close-in proximity will be something like a Theremin. It's not just for conductive things, since anything affecting the local electric field counts; hence, the "radar cross section" (if you will) of a metallic or conductive object will be large (actual size), while that of a dielectric (say, a solid nylon rod, or heavy PVC pipe) will be smaller, but still detectable. That of something with a very low average dielectric constant (foam, thin film?) will be very difficult or impossible to detect, even in direct contact.
Note that gradient matters, so if you put a hunk of dielectric in the close-in field, it will be partially shielded by the expanse of the antenna's field to begin with. Like how supermassive black holes have less shear at the event horizon than small ones (yeah, a familiar, intuitive analogy, I know..). A small antenna is desirable to sense smaller objects at closer distances. (This doesn't apply for large conductive objects, which short out all that field.)
If it's to detect something rigid, it's hard to go wrong with physical contact; it could be the deflection of a lever (e.g., cherry switch, "cat's whisker", ...), or the deflection of an air stream (pressure, direction, turbulence?), or... who knows. It has to apply some force, sooner or later, so could tear very thin film (or get broken itself), and probably will eventually abrade things (e.g., if it's sensing a large solid angle by spinning around quickly, you probably wouldn't want to put your eyeball up close to it..)
Reflection works for reflective objects, but you need a frequency where that's good; microwaves might offer reasonable performance without being especially directive (like optics) while being potentially more sensitive (i.e., having around a 1/4 wave range to pick up some real good con/destructive interference?). Same dielectric limitations would apply.
Doing it optically might be too complicated; simply monitoring diffuse light upon an illuminated sensor and detecting reflection or transmission changes would be subject to pretty strong limitations on the nature of the object (is it white or black, transparent or opaque, and how much of a feel do you get for its "cross section" as a result?), and line-of-sight (even with diffusers). You could get infinitely fancy, like using scanning lasers and such (which are indeed used for spacial 3D scanning heads), with some similar limitations, but by going into the serious power of radar/lidar stuff, that can overcome some of those limitations (e.g., if the return is weak, but we know its distance, that's still telling us exactly where it is; or if there's a hole in one spot, maybe it's reasonable to interpolate from nearby distances).
And if you want to get *really* esoteric, there are more types of radiation than electromagnetic and thermalized neutral baryon ensembles (i.e...poking it with matter). Neutrons are hardly biologically friendly, but have the potential to tell you a whole hell of a lot about the subject of interest (neutron activation and subsequent gamma and beta decays; possible neutron diffraction / scattering). Alphas are big burly ions that bounce around things; maybe useful in short distance (smoke detector?) or vacuum conditions.
As for inexpensive, these latter options certainly aren't going to fit, but it'll be damned hard to beat a ten cent webcam (with suitable software -- might not be inexpensive if NRE is required!), which can capture and
*identify* birds at much greater than "close in" distances. Quite a lot of effort has gone into hiding them; pinhole types are cheap and plentiful, and easy to disguise.
Though there is something morbidly attractive about shooting high energy particle beams at flying things.
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Tim