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I am not sure I understand Bitcoin very well. What I understand is that Bitcoin is there because of the network.
What happens if the network is down for any reason?
What happens if the miner that keeps my records of Bitcoin dies or doesn't want to open his computer again?
Bitcoin network works on a Peer-to-peer mode, so "taking the network down", would require taking at least a good portion of the Internet. Should something like this happen, Bitcoins might not be the biggest of your worries.
A Bitcoin miner stores little to no information about the network itself. It works in a simple terms of fetching data, calculating some hashes and returning results. It does not need much more than that, so miners going offline for awhile wouldn't hurt the network much, asides slowing down block generation for awhile.
A Bitcoin client on the other hand stores all the data from the Bitcoin network. Should it go offline, it still stores all the data on the hard drive and will try to reconnect when it can. Should a good chunk of the network go offline, the blockchain might fork, as each part of the network would try to work only with the data it possesses. Some parts of the network might see only a part of the transactions, while others would see other transactions. When such "islands" would meet, all the clients will try to establish which blockchain is the longest. The shorter blockchain will have all its transactions extracted from it and if there is no double-spend, those transactions might be included in future blocks.
All in all, it is very hard for the Bitcoin network to go down, and the main Bitcoin client is prepared for possible network segmentation.
Nothing happens. The longest chain of blocks in existence, so long as it builds off the latest checkpoint) is always the valid chain, no matter what. So long as at least one node brings it back eventually, there's no problem. (And you can keep it yourself if you're particularly worried about it. If you're running the standard client, you already are.)
Unlike a bank account, your Bitcoin balance doesn't have to exist in any particular place to be valid. The miners create the mathematical relationships that prove it's valid such that anyone at any time can tell that it's valid just by looking at the mined data.
If you connected to any Bitcoin node right now and presented a longer chain of blocks (so long as it included the last checkpoint) than they already had, they would switch over to your chain of blocks and it would become the official chain. It wouldn't matter that prior to introducing it, you had the only copy. It's mathematically valid, so the network would accept it.
Miners add new blocks, but every node keeps a copy of each complete block. There are this many nodes (i.e., tens of thousands):
Once a block is announced to the network, that miner no longer needs to do anything -- that block has already been signed and gets propagated by peers over the network.
