Yes hosting your own domain is not the problem at all. Cost on a monthly basis can even be under $5, maybe more realistic is between $5 and $10. Many have "unlimited storage" and "unlimited bandwidth" (take with a grain of salt, but for personal use it more than likely is correct, you do not even have to create a web page really).

The biggest issue (not really but more complicated) is that you will have to get photo software on your own computer to resize and reduce the photos before uploading. You cannot upload a 5mg photo and expect to share it really. Photobucket did this for you automatically, so you would have to do it yourself. The software is out there, and some of them will do large batches at a time.

Then the only other time problem is creating all of the folders/directories and actually uploading them to the servers.