To make evasi0n work with iOS 7.0.5 one way is to patch 11B511 into 11B601 in the binary. Ehmm btw… the patch are not hex bytes but simple ASCII iOS build numbers that need to be fixed.
If patching binaries isn’t exactly in your repertoire, there are multiple unofficial version of Evasi0n7 jailbreak tools already floating around the web which can jailbreak iOS 7.0.5. While they do work without any issue, it’s best to avoid these unofficial releases and wait for the official version of Evasi0n7. Now that it is confirmed to be working on 7.0.5, it’s more than likely that the Evad3rs team will come through with an updated version of Evasi0n right on cue which supports the latest iOS 7.0.5 release.
MuscleNerd’s tweets seems indicate it probably doesn’t, but its still best to avoid the 7.0.5 update for now.
With iOS 7.1 now a few betas in, it looks as though this will be the next major release across the board, with today’s offering probably being the last of the iOS 7.0.x family. As an advanced reminder, as with any new firmware update, do not go ahead and download / install iOS 7.1 if you wish for your device to continue running Cydia, for it will be patched against the tenacious Evasi0n7 jailbreak tool.