Yes, you can. Simply do the following:
Step 1 — make you profile private

It will disable both — the profile page itself and every link to it.
Step 2 — change your real name to “Somebody” or “Anonymous”, etc

Step 3 — change your profile photo
Go to your “Public Profile — Edit Profile”:

and remove / change your photo.
Step 4 — submit your "Open to a job" post but don't include any private information
Be careful with disclosing private information in the post body.
Step 5 — wait for approval of your post
An admin will contact you and ask to provide an evidence of skills you honed. We need to know that you are indeed professional before approving an “Open to” post from you. Often, it involves taking a look at your job history, freelance exchange / LinkedIn profile. In other words, we have to get acquainted with you before approving your posts, you can NOT remain anonymous for our team.
Step 6 — done
Recruiters can send you messages, but only you decide whether to write back to them.