Currently, even the nearest stars are lightyears away, and impossible to reach in our lifetimes. If space is always expanding, and was once infinitely smaller, then at what point in the past was space so much smaller that the average distance between stars was less than light days? Was there ever such a time?
As the universe expands each individual galaxy stays roughly the same size, with stars on orbits of roughly constant diameter, so the stars within any given galaxy were no closer together a long time ago than they are now (at least as far as cosmic expansion effects are concerned).
The distances between galaxy clusters were smaller in the past, and a good way to get a sense of this is to note that the ratio of distance between them now to distance between them a long time ago is equal to the ratio of wavelengths in the light received and emitted. If we receive light from a galaxy and the light arriving has a wavelength twice as large as when it set out, then the universe was half as small when the light set out (that is, distances between galaxy clusters were then on average half what they now are).
To find a time when galaxies were not many lightyears apart you have to go so far back that you arrive at times before the formation of galaxies, so there never was such a time.
[Added remark in answer to a question in the comments concerning galaxy clusters. One galaxy cluster drifts away from another because the initial conditions gave them velocities of this form. This general condition is called the "Hubble flow" and it leads to the cosmic expansion. It is what things would do if they only experienced the average cosmic gravitation, without any local bumps owing to a non-homogeneous matter distribution such as a galaxy. Meanwhile everything attracts stuff near to it and this can lead to bound groups such as solar systems, galaxies and galaxy clusters. This binding is sufficient to turn the relative velocities around so that each bound group does not drift apart, nor does it expand (unless some other process intervenes).]
Although the distance between stars doesn't really change due to the expansion of space over the evolution of the universe, the region of space around our Sun is quite sparse compared to some places in the universe. As such, your desired conditions could exist not in the remote past but today, just in a different location.
Here's an article which says "In the center of the galaxy, stars are only 0.4–0.04 light-years apart". 0.04 light years is less than 15 light days.
And another article I found with some quick googling claims:
But some galaxies pack stars even tighter. M32, one of the Andromeda Galaxy's satellites, has the highest measured stellar density of any nearby galaxy — around 20 million stars per cubic parsec in its core! Not even HST can resolve M32's core into individual stars. A typical stellar separation at this density works out to 0.008 light-year, or 500 AU — about 12 times the Sun-Pluto distance — between stars.
0.008 light years is just shy of 3 light days.
These figures come from "average" or "typical" distances to nearest neighbour in dense regions, implying that there will be some even closer than that. How much closer is possible/plausible, I couldn't really say.