1. She can run as fast as him.

2. She can run as fast as he.
3. She can run as fast as he can.
4. She can run as fast as he can run.
--------------------
Are they all grammatical? Can we use all of them?

1 is incorrect.

2 and 3 are the most often used.

4 is correct, but wordy and not used much.

5. She can run as fast as he can run fast.

6. She can run fast. He can run fast.
=> She can run as fast as he can run fast.

[Thank you for your help. What about these ones?]

Those are too wordy/repetitive. Here's the best way to say this:

She can run as fast as he can.

1. "She can run as fast as him."

This sentence is commonly used in informal speech and writing, but it is not grammatically correct. Typically, you would use the subject pronoun "he" instead of the object pronoun "him" after the verb "can." Therefore, this sentence should actually be: "She can run as fast as he."

2. "She can run as fast as he."
This sentence is grammatically correct. By using the subject pronoun "he" after the verb "can," it maintains grammatical accuracy and clarity.

3. "She can run as fast as he can."
This sentence is grammatically correct. It uses the comparative form "as fast as" followed by the subject pronoun "he" and the auxiliary verb "can." This construction is commonly used in both spoken and written English.

4. "She can run as fast as he can run."
This sentence is grammatically correct. It uses the comparative form "as fast as" followed by the subject pronoun "he" and the auxiliary verb "can," and then repeats the verb "run." While this construction may sound a bit repetitive, it is still grammatically accurate.

So, out of the four options you provided, the technically correct and grammatical sentences are:
- "She can run as fast as he."
- "She can run as fast as he can."
- "She can run as fast as he can run."