I would look into a Honda Fit. It provides a lot of room and is reliable while maintaining small exterior dimensions. Any year is reliable, but 2015 and newer Fits were nicer because of a redesign.
Another possible candidate is the Kia Soul. You can get a newer one with more features than the Fit because Kias depreciate a lot. Perhaps a Kia dealer will have a CPO Soul in your budget. If so, you'll get a 10 yr 100k mile powertrain warranty. The Soul provides a lot of space like the Soul but feels more substantial and it was available with more features than the Fit. I suggest a 2014 or newer, but any year is reliable.
The Mazda3 Hatchback handles well, is reliable, and you can fit a surprising amount of stuff in the trunk despite having a tight interior. Look for 2012 or newer ones.
Since you need space, I would avoid sedans. Hatchbacks and wagons have the best ratio of more interior space and smaller exterior. SUVs often provide more space than hatchbacks, but they are much larger on the exterior. For example, a Mazda CX-5 may have 20% more interior space than the Mazda3, but it may be 40% bigger. Just something to think about if you want something that handles well or is easy to park.