can I take full custody of my friends baby?

It doesn't work like that. 

To adopt her baby, first the baby needs to be free and clear for adoption.  Meaning both parents need to relinquish their parental rights and the state only allows a parent to relinquish their parental rights if it is so another person can adopt the child and that person is already approved to do so.  The father needs to consent to this too.  An unknown father or not putting the father on the birth certificate makes adoption harder, not easier.  If the father is not listed or is unknown, the state will require a reasonable effort be made to locate the father and any potential fathers are DNA tested to be sure they are the father of the child.  When/if there is a match he will be given the opportunity to obtain his parental rights and parent the child.  The timeline here varies state to state but it can be a couple years.

Every adoption type in the U.S. requires a home study.  Home Studies, a home study has a cost associated with it and averages $1000-3000.  For private adoptions where the prospective adoptive parent is known to the expectant parents in some cases they will accept a home study from foster care which is free, but not always.  A home study is extensive and instrusive.  They will require fingerprints, a criminal background check, interviews with you, 3-5 references, a physical from a physician and a financial disclosure.  If you have other adults living in your home, they also will be required to be on the home study even if they are not adopting.  They will go through the fingerprinting and criminal background check and they will be interviewed too.  If you already have kids, they will be interviewed too (age appropriate questions).  A home study can take several months to complete.  If you are receiving assistance, that means you are not financially stable and are unlikely to actually pass the home study.

Adoptions require 2 attorney's in most states.  One who represents the expectant parent(s) and one who represents the prospective adoptive parent(s).  The prospective adoptive parent(s) is responsible for all legal fees (meaning you pay for both).  A private adoption can cost around $5000 or so.

In some states the adoptive parents are required to pay some of the birth and prenatal care expenses if the mother cannot.

After you have paid all of this, the expectant mother can change her mind when she gives birth and if she does..  you are out that money.  She is not required to reimburse any expenses.

There is also a revocation period.  Revocation can be anywhere from 3-30 days in most states and sometimes it's up until adoption finalization which in most states is about 6 months.  Revocation means she can take the baby back during the revocation period.

If somehow this was done without the fathers consent and he later finds out, he can have the adoption overturned in court.

Having guardianship is a little different, but the home study process is similar.  She may be able to give you legal guardianship but we aware she can take the baby back too with guardianship and they are more likely to give kinship guardianship to a relative if they are interested.

You probably would not be approved for adoption given that you are currently unable to provide for yourself. It would be in the child's best interest to be placed with a stable family, and to not be disrupted by a birth mother popping in and out of their life as your friend probably intends to do by wanting to give you custody. Although your friend might like the idea of visiting her child whenever she wants to, it is extremely confusing and disruptive to a child, and is likely to compound feelings of abandonment. Not healthy at all.

During the homestudy, they will look over your finances. Once they see you are on government assistance I can guarantee that you will be disqualified from adopting the baby. If you can’t even provide for yourself without help, how are you going to provide for another person? Adopting costs thousands of dollars that you, quite frankly, don’t have. Sorry.

Can you afford to raise the child? Government assistance doesn't pay much. Seriously assess the costs before you commit to taking on this child.

It is unlikely you would be approved to adopt, but she might be able to give you legal guardianship.

https://www.acf.hhs.gov/cb/faq/custody3

No.  You, like anyone else, must apply to be an adoptive parent or legal guardian.  These requirements include: 

Must be a legal adult;

Must be physically able to fulfill the responsibilities;

Must have ample time to care for your children; and.

Must be able to afford to raise your children, via their own income or via assets you leave for the children's care.

If you do not meet those requirements, you won't be granted custody of any kind. 

I'll just consult my crystal ball