Obviously most dead tissue things rots, so you need sources that have been preserved, e.g. mummies or mammoths in permafrost
Then it is chemically degraded over time. Split into short sequences. However PCR, the polymerase chain reaction means in theory you can amplify just one molecule into enough DNA to analyse. So even with a fragmented sample, by overlaps between sequences you may be able to reconstruct quite a long sequence. Among the problems there are contamination by the much more abundant DNA of all sorts of things like bacteria and the humans who have handled the tissues.
DNA sequences of samples from thousands of years ago have been reconstructed. For example from Neanderthals it was fairly recently announced. But when you read of such
tours de force remember that the sequences are
mitochondrial DNA, that is non-nuclear DNA which is several orders of magnitude more abundant (by molarity) than nuclear DNA. Which is fine if your purpose is to say something of evolutionary relationships, e.g. the relationship of Neanderthals with present human population. It will in any case always be related - for great part identical - to a known DNA and you know what you are looking for.
For mammoths getting the material is fairly favourable. The elephant genome is fairly completely known and checking now I am amazed to discover that so is the mammoth!
http://www.nature.com/nature/journal/v456/n7220/full/456330a.html The power of present techniques is fantastic. The implantation of a fertilised mammoth egg in an elephant I would guess could work - they are closely related. As experimental sciences go it is not for the impatient since an elephant has a 3-yr gestation. But knowing the DNA sequence is not the same thing as having a mammoth cell. I doubt they have got any mammoth cells they could resuscitate. Barring future breakthroughs I guess they would substitute the relatively few genes that are different in the two species. But that is still a lot of genes. So probably they wouldn't change all of them but just some crucial ones, to get something that wasn't a mammoth that could be related to any mammoth that ever lived but something that looks and functions quite like one. They won't know which of all the genes the crucial ones are though they might work out some that are important.
It would need quite a lot of determination or else further breakthroughs; I wouldn't bet you will see it in your lifetime but you may just see it for some easier species.