The USB-SATA bridge firmware in Seagate's 3TB GoFlex externals is configured with 4KB LBAs. This is to allow them to be compatible with Windows XP. If you remove the drive from its enclosure and attach it directly to a SATA port on your computer's motherboard, then you will be exposing the drive's 512e LBAs. Sector 0, where the partition table resides, will still be in the same place, but every other sector location will be out by a factor of 8.
The actual capacity is ...
349.31 x 8 = 2794.48 GB
In short, you need to reinstall the drive in its enclosure, or find another enclosure that defaults to a 4096-byte sector size.
"Initialisation" is data destructive. It deletes the entries in the partition table.
That said, the error message is implying that Disk Management is unable to write to sector 0. Is Disk Management now reporting the correct size for the physical drive? Does it detect an existing 3TB NTFS partition?
The fact that you can access sector 0 when the drive is outside the case but not when it is inside would suggest that there is something wrong with the case, or perhaps there is some kind of security issue (password, write protection, access rights).
Disk Management appears to be correctly detecting the drive behind the USB-SATA bridge board inside the enclosure.