The good news is that you will be able to easily find acreage within 30 minutes of Burnsville (15 minutes or less even). There are some homes for rent not far from Burnsville but I don't have the number for them. One is one side of a duplex that has a fenced in yard. The back yard runs along 160th street (in Apple Valley-right next door to Burnsville). It would be about a 5 minute commute-depending on where in Burnsville. I don't know if they take dogs. I don't have a phone number but if you are driving in the area they have a sign out. Most homes for rent are pretty much advertised like that around here.
Here is a link to some homes for rent
Thisweek Live: Thisweek Newspapers/Dakota County Tribune - Classifieds
This newspaper would be a good resource for you for that area. All of these homes are about 15 minutes or less to Burnsville.
Pretty much anything in Apple Valley is going to be fine. There are some not so nice areas in Burnsville but you will be able to spot them just driving through. Savage is close to Burnsville but kind of dumpy. There are some nice neighborhoods there.
Go south a bit to Lakeville or even over to Rosemount. Again, all of these are 15 minutes or so to Burnsville and will have everything you want-good neighborhoods, lots of parks, etc.