Not sure about what you actually mean by reaching Mesaieed beach "without a car", but based on what i get to understand, I think you're talking about being able to drive through the sand to get to the beach....right? The surface area around the beach is faily hardened and you'd be able to drive any car to within a few meters from the waterline. There is a gravel road along the beach and you car park your car there.

Alternatively, are you saying you don't have a car and you wanna goto Mesaieed beach from Doha? Well....in that case, i donno what to say to you.

Mesaieed beach is fairly public and you cannot wear bikinis there. You must select some other beaches like Dukhan or Shamaal or Ruwais to wear skimpy outfits coz these beaches are desolate.

I hope that answers your question.