How to base work of my Agile team on issues of different types (not on the Stories)?

[Oleksandr Sulzhenko]

The Board displays rectangles of type Story, not a Task (and don't even mention a Sub-tasks Jira just can't).

Is Jira capable of it?

Why Jira tie me to Stories? What if I would like to put into my sprint: 3 stories, 5 tasks, 10 epics and 32 sub-tasks all the things not related to each other (e.g. there's no a task among that 5 that belongs to a story among that 3)?

[Nic Brough -Adaptavist-]

You don't put epics or sub-tasks into sprints  (Well, technically, you do with sub-tasks, but it's indirect - they are in a sprint because they are part of a parent issue which you take into a sprint)

All issues at the issue level will go into a backlog and hence a sprint, the name of the issue type is irrelevant.  My guess is that your "tasks" are a type of sub-task.  If you want to sprint on tasks, then you'll need an issue type named "task" as a standard issue type (no a sub task type)