What am I allowed to bring in US Embassy while doing passport renewal 

The embassy Security Regulations page isn't 100% clear.  It states "You will NOT be allowed to enter the U.S. Embassy for your appointment carrying any of the following items:

• Any oversize shoulder bags or purses.
• Any oversize handbags and travel bags, large back-packs, briefcases, or suitcases.
• Applicants can only carry in clear bags containing papers needed for their appointment "

So can a normal size bag [what is considered normal?] be taken in or is it only the clear bag containing papers?

Is there a facility to store bags at or near the embassy?

The restriction means suitcase size bags.

I have been allowed to enter with what might be called a man bag without a problem.

Pay attention to the electronic items restriction. I forgot my pen drive one time and had to leave it with my phone.

The will let you leave a carry on sized bag at the security check area.

 

I have seen a briefcase taken in, or like ubonjoe said "a man bag". I always carry a large book size, hard, clear, plastic thingy thing.

Women can bring in their handbags, no problem.  I've also brought in small shopping bags. But no phones or other electronics.

UPDATE: I brought a smaller than 'normal' size back pack to hold a change of clothes, toiletries and my paperwork. Arrived 30 minutes early at the consulate and was allowed to enter. Phone was taken and backpack went through the security scanner. I was then asked for an additional ID. All of this was taken and I was given a bracelet. I then had to ask for my backpack back in order to retrieve my paperwork.  All was then stowed in a 30cm x 30cm and maybe 45cm deep bin. On exit I surrendered my bracelet in exchange for my backpack and ID.  All in all for my 9am appointment I was done by 9:15 so just 45 mins in and out.

Another option is to make use of ACS outreach (where they don't care what you bring in and the lines are minimal especially if you're staying there).   Just make it a part of a mini holiday and book a room at whatever hotel they are using for the particular outreach.   

Remember that at the Outreach visits they no longer accept cash for services.