Redmine

You need uninstall conflicted versiones

gem uninstall rake -v=0.9.2
gem uninstall rails -v=2.3.12
gem uninstall rack -v=1.1.2

Check compatibility notes: http://www.redmine.org/projects/redmine/wiki/RedmineInstall

I cannot uninstall them since those are system packages installed by the host.

I'm trying to install Redmine in shared hosting environment.

Ok...

Uninstall all system packages except rubygems (so called in Debian) and install rake, rails ans rack with gem directly, gem is a package manager for ruby like dpkg for Debian.

Remember install gems in this order and versions

:~# gem install rack -v=1.1.1
:~# gem install rake -v=0.8.7
:~# gem install i18n -v=0.4.2
:~# gem install rails -v=2.3.11

:~# gem list

*** LOCAL GEMS ***

actionmailer (2.3.11)
actionpack (2.3.11)
activerecord (2.3.11)
activeresource (2.3.11)
activesupport (2.3.11)
i18n (0.4.2)
rack (1.1.1)
rails (2.3.11)
rake (0.8.7)
ruby-ole (1.2.11.2)
spreadsheet (0.6.5.9)

I'm reluctant to do so, since according to forums.cpanel.net/f5/ruby-rails-issue-222852.html#post917402 cPanel requires rails 2.3.12 and I wouldn't want to damage the cPanel (it's a bit more important than redmine...)
Also, someone told me that

it's probably better to avoid mucking with system packages.

Since I'm no expert in ruby, I don't want to do some steps that will damage my account.

I tried installing RVM but this fails too since there is no git, and after gem install git it cannot be executed (no executable is found anywhere on the system).

ok

You can try

rake _0.8.7_ db:migrate**************** 

It is the same result as before:

/usr/lib/ruby/site_ruby/1.8/rubygems/dependency.rb:247:in `to_specs': Could not find rake (= 0.8.7) amongst [actionmailer-2.3.12, actionpack-2.3.12, activerecord-2.3.12, activeresource-2.3.12, activesupport-2.3.12, cgi_multipart_eof_fix-2.5.0, daemons-1.1.4, fastthread-1.0.7, gem_plugin-0.2.3, mongrel-1.1.5, rack-1.1.2, rails-2.3.12, rake-0.9.2, rubygems-update-1.8.7, sqlite3-1.3.4] (Gem::LoadError)
        from /usr/lib/ruby/site_ruby/1.8/rubygems/dependency.rb:256:in `to_spec'
        from /usr/lib/ruby/site_ruby/1.8/rubygems.rb:1200:in `gem'
        from /usr/bin/rake:18