How to get something from the state / store inside a redux-saga function?

This is what "selector" functions are for. You pass them the entire state tree, and they return some piece of the state. The code that calls the selector doesn't need to know where in the state that data was, just that it was returned. See http://redux.js.org/docs/recipes/ComputingDerivedData.html for some examples.

Within a saga, the select() API can be used to execute a selector.

As @markerikson already says, redux-saga exposes a very useful API select() to invoke a selector on the state for getting some part of it available inside the saga.

For your example a simple implementation could be:

/*
 * Selector. The query depends by the state shape
 */
export const getProject = (state) => state.project

// Saga
export function* saveProjectTask() {
  while(true) {
    yield take(SAVE_PROJECT);
    let project = yield select(getProject); // <-- get the project
    yield call(fetch, '/api/project', { body: project, method: 'PUT' });
    yield put({type: SAVE_PROJECT_SUCCESS});
  }
}

In addition to the suggested doc by @markerikson, there is a very good video tutorial by D. Abramov which explains how to use selectors with Redux. Check also this interesting thread on Twitter.

I used an eventChannel to dispatch an action from a callback within the generator function

import {eventChannel} from 'redux-saga';
import {call, take} from 'redux-saga/effects';

function createEventChannel(setEmitter) {
    return eventChannel(emitter => {
        setEmitter(emitter)
        return () => {

        }
      }
    )
}

function* YourSaga(){
    let emitter;
    const internalEvents = yield call(createEventChannel, em => emitter = em)

    const scopedCallback = () => {
        emitter({type, payload})
    }

    while(true){
        const action = yield take(internalEvents)
        yield put(action)
    }
}