The problem is not how close to the edge your printer can print. A 3x5
card is so small that most printers can't properly pick up and feed it
through the printer without jamming. Some photo printers are designed
to print on small paper sizes. If your printer can handle small paper
sizes, you should be able to select the 3x5 size in Word by clicking on
File | Page Setup | Paper | Paper Size.
I suppose you could print 4 cards on one letter-size page, and then
manually cut them into individual cards. For this, you can create a
landscape page layout with 0.5" margins on all sides, and insert a
two-column by two-row table with the row height set to exactly 3" and
column width set to exactly 5".