I use Dropbox.
It basically works as a folder on your computer. It automatically backs up all the files in it online when you put them in the Dropbox folder on your computer.
If you put them in the public folder, you can then right click on the file and select 'Copy public Link' and that is the URL you use to post on forums, the same as you would do with imgur.
It's a very simple thing to use, very reliable, and very popular, especially with businesses. Of course you also have the benefit of logging into the website and uploading/dowloading files to/from your dropbox folder if you're away from your own computer. I even add files to my Dropbox directly from my phone.
To be honest these days it's very difficult to 'lose' data held by the likes of Dropbox as everything is backed up in different physical locations - even if one Dropbox data centre burnt down today, it wouldn't matter so much, as the content is backed up elsewhere. Also with Dropbox you have the added security of the files still being on your computer too. If your hard drive went, you could buy a new computer, install dropbox again and your content would be downloaded to the new computer.
I use Dropbox Pro, which is well worth it for 1TB of storage. However if you're only using Dropbox for hosting images on Mayhem etc then you can get away with using the free version - You still get a few gigabytes to play with.