It seems ClimateTools insists on old versions for a lot of packages (e.g. check out the large amount of unmerged CompatBot PRs Pull requests · JuliaClimate/ClimateTools.jl · GitHub )
One way around the issue for me was to start by pinning MultivariateStats and StatsBase to their newest versions and to install ClimateTools after that:
(climetoolstest) pkg> st
Status `/tmp/climetoolstest/Project.toml`
⌃ [4f4ee721] ClimateTools v0.13.0
[6f286f6a] MultivariateStats v0.10.0
[2913bbd2] StatsBase v0.33.21
Info Packages marked with ⌃ have new versions available
That is a pretty ancient version of ClimateTools though. Everything precompiles fine now (but keep in mind that Project.toml is not enough to fully define the system, you need Manifest.toml for that, because the order of installations might matter to the resolver).
My amateur guess is that somewhere deep in the mutual dependencies of these packages ClimateTools needs an old version of a package, which is listed as supported in the Project.toml file of MultivariateStats (or dependencies), but is not actually supported. Someone has release a new version of their package and has not noticed that they rely on a newer feature in one of their dependencies. That should be found and fixed, but I think it is much more urgent that someone goes through the CompatBot pull requests for ClimateTools.