You can certainly add a memory functionality of some sort. You could even store the data in EEPROM so it'll survive power cycling. You'll need a way to control it of course. Maybe menus shown by the LCD.
As to whether the Nano can do the job, pins will probably be a constraint. Some for the LCD, more for the keypad, two per stepper and one per limit switch. If you're going to add encoders or some other mechanism to detect movement, you're likely to run out. The Uno doesn't have any more, so you may need a Mega. You can prototype with the Nano of course while you figure out how many pins and how much memory you will need.