How to achieve uptime of the website?

Cloud hosting from Amazon. Create instances in different parts of the light and forward.

Write the causes of falls.

GSLB cluster.

And how can you answer the question "... in case of problems with the current hosting", if the name of the current hosting is NOT NAMED?..
Habr — not a bunch of mediums and psychics...
Suggest on the subject of the health of your site to talk solely with the current host...

When talking about very large money then I would recommend to use the services www.akamai.com expensive, but effective.
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If you look for something much cheaper you have to rent servers in different data centers and connect them.
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For users which countries oriented site?
If Europe's data centers in Germany, Holland, Sweden, Norway (Teliasonera and The bunker).
If we are talking about America then it's likely Amazon and Verizon — of course they are suitable for Europe.

Reliability can be increased to infinity and it will not protect against all cases. And Yandex, and Facebook fell.
To give advice, you need to understand the architecture of the site. Causes of drops too many and each can be protected. It all depends on the money. Can be affected from force majeure (burned DTS), and from various attacks.
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Means uptime'a
RAID
Database replication, and database sharding
Balancer (either in nginx or DNS IP) load N frotando.
Post frotando and backendu for datacenters
BGP
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All the money may not be enough :)

scatter wdsci, to by instances in various parts of the world... about syncing, then who will think? And DNS who would the shaman, if you fall something? don't forget that they only work as a round-robin.
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2 servers in the M9 and M9-ENEA, heartbeat, smart gateway, which will respond and transmit the IP address to the second server if the first one falls. Value — 9-15K+rent/colo server+traffic.