Amazon Web Services
The Tarsnap service is built on top of the solid platform provided by Amazon Web Services. Data archived via Tarsnap is stored to the Amazon S3 storage service (the original version, which can survive the loss of 2 datacenters, not the "reduced redundancy" version which can only survive the loss of a single datacenter) before it is acknowledged by the Tarsnap server to Tarsnap clients, so the probability of Tarsnap data being lost due to hardware failures is extremely low — somewhere approaching the probability of a nuclear war or meteor hit destroying most of the US.
The Tarsnap server resides in the Amazon EC2 compute cloud, and at the present time it is possible — but quite unlikely — that a hardware failure would result in the Tarsnap service becoming unavailable until a new EC2 instance can be launched and the Tarsnap server code can be restarted (note that this would make data archived via Tarsnap unavailable, but would not cause it to be permanently lost). So far such an outage has never occurred; but over time Tarsnap will become more tolerant of failures in order to minimize the probability that such an outage occurs in the future.
For more information (including technical details about constructing a log-structured filesystem using S3 as back-end storage, caching metadata in EC2, and bundling writes in order to save on S3 per-request costs), see the author's blog post from December 2008, How Tarsnap uses Amazon Web Services.