Eastham has arrived on a one-month loan from Championship club Blackpool while former Leicester and Derby defender Stewart was a free agent after leaving Millwall in the summer and has signed a short-term deal with Keith Curle's side.
Eastham, 21, has experience of League One football having appeared 27 times for Bury during a loan spell last season. The centre-back has also enjoyed loan stints at Cheltenham, Carlisle and, most recently, Fleetwood.
Stewart, 30, has been training with Notts and has impressed Curle enough to earn a contract.
Both players are available for selection for Saturday's league match at home to table-topping Tranmere.