ONE HOUR ANYWHERE NEAR IT AND YOU'RE DEAD
This claim is probably too broad and the information is outdated, but the general idea is correct. A quick search revealed a few relevant details:
From Atlas Obscura:
According to the Natural Resources Defense Council, the lake is so polluted that its waters will kill a human within an hour.
From Basement Geographer (which is referencing a broken link):
spending just an hour on its shores would result in certain death
Also, from the Wikipedia article
The radiation level in the region near where radioactive effluent is discharged into the lake was 600 röntgens per hour (approximately 6 Sv/h) in 1990
According to Basement Geographer, the last reactor in the area closed in 1990, so the radiation will only have gone down since then. Also, I didn't see any information as to the radiation levels any place other than where the radioactive waste enters the lake. As such, I don't know what the current levels of radiation are at various places around and in the lake.
Something else to consider is how good water is at stopping radiation. According to the XKCD What-If blog,
For the kinds of radiation coming off spent nuclear fuel, every 7 centimeters of water cuts the amount of radiation in half.
Of course, Randall looked at a scenario where the radioactive material is not mixed with the water. In Lake Karachay the radioactive material is not contained, but the radiation from the material at or near the bottom of the lake would not reach you unless you were underwater. If all (or at least most) of the radioactive material has settled to the bottom in the decades since they stopped dumping waste into the lake, then it might actually be safe to float in the middle of the lake (getting past the radioactive shores would still be a problem, however).
So other than going to Lake Karachay with a geiger counter, we can't know for sure whether or not there are areas where the radiation levels are safe. There is a lot of radioactive material there, so it is almost certain that there are parts of the lake or the area around it where you could receive a lethal dose of radiation within hours.
In any case, the lake in the picture looks like a fine place to visit - the healthy trees, the size of the lake, and the island (see Basement Geographer for satellite image of the lake) make it obvious that that isn't a picture of Lake Karachay. It is actually Phantom Ship, an island in Crater Lake, Oregon, USA. [Hat tip: @ChrisW]. This photo of the lake appears to be the original (or at least derived from the same source):