2011 iMac with Dead internal HDD. Possible to boot from an external disk?
There is no hard and fast rule for this. Sometimes when a HDD fails to boot, you can boot from another attached bootable disk. Sometimes, however, the HDD fails in a way it won't permit booting from attached drives because it seems to monopolize the bus.

You don't have USB 3.0 or firewire on this, even if it is a late 2011 iMac core i7. So you'd be limited to Thunderbolt or USB 2.0, which is going to be a little slow. 

I know iMacs are hard to take apart, but your best bet is to look at the instructions or video on how to install an internal SSD from Other World Computing (MacSales.com) or, iFixit.com. If you think this is possible, it's a cost effective way to really fix your iMac and make it a lot faster. 

edit: I wanted to add that USB 2.0 external HDDs are cheap and slow, Thunderbolt drives are expensive, and you can now get a decent SSD to install internally within your budget.