Can a smartphone (android) work like an iPod Touch using only WiFi without a SIM card?

I'm trying to find a cheaper alternative to an iPod Touch, so I wondered if I could purchase a used android based smartphone and not install a SIM card? Would I still be able to use the smartphone via WiFi to surf the internet, check email, play mp3s, play games, etc., but not use it as a phone or use cellular signals? Yes, I could purchase a 10" or 7" tablet computer for ~$250, but I want something even cheaper that could easily fit in my pocket.