How to get multiple mutable references to elements in a Vec?

This is possible using safe Rust as long as you're very careful. The trick is to leverage the unsafe Rust code in the standard library behind the safe .iter_mut() and .nth() methods on Vec. Here's a working example with comments explaining the code in context:

fn main() {
    let mut data = Data::new(vec![2, 3, 4]);

    // this works
    let slice = data.get_mut_slice(1);
    slice[2] = 5.0;
    println!("{:?}", data);

    // and now this works too!
    let mut slices = data.get_mut_slices(vec![0, 1]);
    slices[0][0] = 2.0;
    slices[1][0] = 3.0;
    println!("{:?}", data);
}

#[derive(Debug)]
struct Data {
    data: Vec<Vec<f64>>,
}

impl Data {
    fn new(lengths: Vec<usize>) -> Data {
        Data {
            data: lengths.iter().map(|n| vec![0_f64; *n]).collect(),
        }
    }

    fn get_mut_slice(&mut self, index: usize) -> &mut [f64] {
        &mut self.data[index][..]
    }

    // now works!
    fn get_mut_slices(&mut self, mut indexes: Vec<usize>) -> Vec<&mut [f64]> {
        // sort indexes for easier processing
        indexes.sort();
        let index_len = indexes.len();

        // early return for edge case
        if index_len == 0 {
            return Vec::new();
        }

        // check that the largest index is in bounds
        let max_index = indexes[index_len - 1];
        if max_index > self.data.len() {
            panic!("{} index is out of bounds of data", max_index);
        }

        // check that we have no overlapping indexes
        indexes.dedup();
        let uniq_index_len = indexes.len();
        if index_len != uniq_index_len {
            panic!("cannot return aliased mut refs to overlapping indexes");
        }

        // leverage the unsafe code that's written in the standard library
        // to safely get multiple unique disjoint mutable references
        // out of the Vec
        let mut mut_slices_iter = self.data.iter_mut();
        let mut mut_slices = Vec::with_capacity(index_len);
        let mut last_index = 0;
        for curr_index in indexes {
            mut_slices.push(
                mut_slices_iter
                    .nth(curr_index - last_index)
                    .unwrap()
                    .as_mut_slice(),
            );
            last_index = curr_index;
        }

        // return results
        mut_slices
    }
}

playground

What I believe I learned is that the Rust compiler demands an iterator in this situation because that is the only way it can know that each mut slice comes from a different vector.

The compiler doesn't know that actually. All it knows is that the iterator returns mut references. The underlying implementation uses unsafe Rust but the method iter_mut() itself is safe because the implementation guarantees to only emit each mut ref once and that all of the mut refs are unique.

Would the compiler complain if another mut_slices_iter was created in the for loop (which could grab the same data twice)?

Yes. Calling iter_mut() on Vec mutably borrows it and overlapping mutable borrows of the same data are against Rust's ownership rules, so you can't call iter_mut() twice in the same scope (unless the iterator returned by the first call is dropped before the second call).

Also am I right that the .nth method will call next() n times so it is ultimately theta(n) on the first axis?

Not quite. That's the default implementation for nth BUT the iterator returned by calling iter_mut() on Vec uses its own custom implementation and it seems to skip past items in the iterator without calling next() so it should be as fast as if you were just regularly indexing into the Vec, i.e. getting 3 randomly indexed items using .nth() would be as fast on an iterator of 10000 items as it would be on an iterator of 10 items, although this is only specifically true for iterators created from collections which support fast random access like Vecs.

If you want unique indexes, which actually makes more sense since you can't/shoudn't have two mutable references to the same element. You can use a HashSet instead of a Vec and use some iterator combinations:

    fn get_mut_slices(&mut self, indexes: HashSet<usize>) -> Vec<&mut [f64]> {
        self.data
            .iter_mut()
            .enumerate()
            .filter(|(i, _)| indexes.contains(i))
            .map(|(_, e)| e.as_mut_slice())
            .collect()
    }

Playground

You can still use Vec for this option, but it would be much more inefficient when using contains:

    fn get_mut_slices(&mut self, indexes: Vec<usize>) -> Vec<&mut [f64]> {
        self.data
            .iter_mut()
            .enumerate()
            .filter(|(i, _)| indexes.contains(i))
            .map(|(_, e)| e.as_mut_slice())
            .collect()
    }