'Run off into the Blue'

Hello,

I am writing on behalf a group of Tolkien fans from Brazil.
Our books are translated to portuguese, and there are expressions that we have difficulty to understand them.

Please, can you help us?

In chapter 'The Shadow of the Past' :
'But in the meantime, the general opinion in the neighbourhood was that Bilbo, who had always been rather cracked, had at last gone quite mad, and had run off into the Blue. There he had undoubtedly and come to a tragic, but hardly an untimely, end.'


What 'Blue' means there?

Some people think that this 'Blue' means 'Sea". Others think that 'run off into the Blue' mean 'go so far away', 'to desappear'.

Coud you explain what Tolkien means exactly?

I apologize and hope that it has not caused any inconvenience.

Thak you for your patience and undertanding.

Sorry, my english is not very good.