By allowing a little error, you can get away with a limited length filter, and it will introduce less error in the frequency domain than sampling the signal directly without filtering first.
Here is an illustrative only example using finite impulse response (FIR) windowed sinc filtering, which is compared to sinc filtering and direct sampling of a rectangular pulse. The length of the rectangular pulse is 10, the cutoff frequency of the sinc is $\pi$. The window function is Blackman–Nuttall and is of length 12 sampling periods at the critical sampling frequency of the sinc.

Figure 1. Sinc (red) and windowed sinc (blue) filter impulse response.

Figure 2. Rectangular pulse (grey), filtered with the sinc (red) and with the windowed sinc (blue) filter. Both ways of filtering introduce time-domain error in the pulse.

Figure 3. Power spectra of the rectangular pulse (grey) filtered with the sinc (red, goes to $-\infty\text{ dB}$ at $\omega = \pi$) and with the windowed sinc (blue) filter. The vertical axis in decibels.
Sinc filter gives no frequency domain error at all for frequencies below $\pi$. The windowed sinc filter gives very little error at frequencies below about 2 (pass band) and has a very high attenuation at frequencies above about 5 (stop band).
Let's consider that the signal of interest is below frequency 2 and we sample the windowed sinc filtered signal at a sampling frequency of 7. The frequencies between 3.5 and 5 will alias to between frequencies 3.5 and 2. Only the greatly attenuated stop band aliases to the band of interest, introducing very little error. So in this case we could get away with a limited length filter and a limited sampling frequency. If we were to sample the rectangular pulse directly, there would be plenty of aliasing to the band of interest.
To continue with a different aspect, consider that we want to measure the start time of the pulse. If we don't low-pass filter, all start times between the time of a sample and the time of the next sample will result in identical sampled signals. If we low-pass filter, the start time can be resolved at arbitrary accuracy by increasing the filter length as the error in the measured start time will approach zero when the FIR filter length approaches infinity. So it pays of to low-pass filter even if the filter is not perfect.