First of all, congratulations on taking up running and I wish you lots of happy miles.

No problem with setting goals, as long as you train properly for them and do not overdo it. Running can sometimes be tough (specially at the begining), so you should focus on enjoying your runs at first.

Once you have achieved that, improved your fitness level a little and are more confident with your running skills, there are many training plans for HM out there, from walking to elite, a quick search led me to Hal Highdon's plans which might be useful for you.

http://halhigdon.com/training/51130/Half-Marathon-Training-Guide

So yes, it is possible, but it will take some training, a positive attitude and lots and lots of power of will to follow the training plan that you choose.