If the vibes were right, then yes absolutely. I'm big about vibes and impressions, so as long as I didn't get skeezed out, I would love to have a stranger stay with me.
I think that there's nothing better in life than the unexpected and spontaneous. There are so many great things that could come from a situation like this--you could make a new life-long friend, or maybe just know a person for a few weeks but have amazing conversations and insights into their lives. Maybe one day, years down the road, your fortunes are reversed and you're the one who needs a place to stay in a rough patch.
You never know. Closing doors before even peeking inside of them is a foolish thing. A stranger is just a friend you haven't met yet.