Why do we not use square filters without windows?
As robert mentioned: Please use the terms convolution and window formally correct in your question!

There's no problem with that aside from the fact that we can't take an infinitely long Fourier transform within finite time!

That's not really a problem, though – through clever overlapping and using the linearity properties of the (discrete) Fourier transform, we can still get valid results that are identical by converting parts of the signal to the frequency domain, multiplying them with the pre-transformed filter, and then converting back to time domain.

You need to ensure that the DFT size and overlap are appropriate, but if you do that, you end up with the well-known overlap-add or overlap-save methods of convolution.

These methods are widely used in DSP! For example, I know for a fact that some hardware / FPGA-base channel emulators and simulators do in fact use a 1024-point FFT, the overlap-add method and a 1024-IFFT.

On general purpose computing hardware (read: PCs and similar computers), FFT filters following that principle are very handy, as they are much more efficient then convolution in time domain. 

For example GNU Radio has such filters and they are blazingly fast. Think of dozens to hundreds of million complex-valued samples convolved with multiple-thousand-tap filters!



Since you're a physicist, you might also be interested in the analog side of things: There's actually ways to physically implement or at least approximate a Fourier transform, and those are used for exactly these kind of things. For example, lenses all have a Fourier transform property!