DNA extraction from agarose gel
Yes, this is possible (and I have done it uncounted number of times) - the method is called "Freeze and squeeze". What you basically do is to run the gel, cut out the band of interest (be careful with the UV light, it causes damage to your DNA and also sunburns, so wear appropriate shielding for your face), dissolve it in a buffer, then freeze it in liquid nitrogen (or the -70°C, doesn't matter) and centrifuge it at room temperature for 10 minutes. During this time the ice melts and the agarose pellets at the bottom. Take the supernatant, do a ethanol precipitation (with glycogen when you only have little DNA) and you have nice and clean DNA. No need for expensive kits and is done approximately in 2 hours.

You can find a very nice and detailed protocol here: "Elution of DNA from Agarose Gels" (The rest of the handbook is also very useful).