Can we or the universe ever achieve
This is not physics.
Physics is a science that tries to describe observable phenomena (using mathematical models). It cannot answers questions like "is
X possible"; you'd have to reformulate the question into "in what kind of situation or system could
X happen".
Because physics only involves
observable phenomena, there is no such thing as "perfect" or "exact"; it is always limited by our ability to observe, to measure the phenomena.
The stated question is like asking a mathematician, "If you had to specify a real number for \$\lim_{x \to \pm0} \frac{1}{x}\$, what would it be?". The expression evaluates to positive or negative infinity depending on the direction you approach zero from, so assigning any real number would be an error.
As mentioned by several others above, we aren't even certain if the electron charge is a constant (the same everywhere and unchanging as time progresses). Thus, "absolutely constant DC current" (even if considered only in some limited time interval) needs to be specified with respect to something, for example as the rate of one-electron charges passing through a surface.
In case you're tempted to insist and play along that definition: even if we found some phenomena where a steady beam of electrons at a specific energy (velocity) is emitted from some source, changing your velocity with respect to the source will change the observed current in the electron beam. Any change in the gravitational potential will change the observed current in the electron beam, too. If you examine the system at shorter and shorter timescales or currents, you'll notice the current (and the related electromagnetic fields) being quantized.
Thus, the only non-nitpicking/language-lawyer answer one can give to the stated question is "According to our current understanding, no: perfectly direct current is impossible, for example due to quantization of current and electromagnetic fields alone. At sufficiently short time intervals, there is always at least some ripple or noise due to quantum mechanics."