Filter implementation and alternatives
Only rational transfer functions can be implemented by a combination of adders, multipliers, and delay elements. The transfer function corresponding to this impulse response (i.e., its $\mathcal{Z}$-transform) is not rational (see
link).

Note that filters with infinitely long impulse responses can be implemented by recursion, so the infinite length by itself is no problem, as long as the transfer function is rational. A way to approximate this impulse response by a realizable filter is to truncate the impulse response by multiplying it with some (right-sided) window function, resulting in a finite impulse response (FIR) filter, which can always be implemented.