Don't men and women meet for coffee somewhere anymore? You meet each other somewhere and sit and talk and share a light supper OR drink coffee and perhaps have a pastry. (Okay, a bit much, but don't break out the shovel just yet)
THEN, if things go well, maybe discuss going to a movie or a concert together or meeting for another non-date; keep the first two or three non-dates public, and if things seem to be going well, maybe one of the two invites the other over to watch a movie......with no expectations of sex or spending the night or anything like that.
Don't men and women do THAT anymore either?