Since Ciutat Vella is a large place which consists of a lot of neighborhoods and districts in the old city of Barcelona, it does not have a special character, every part of the neighborhood has different characteristics.
The Gothic Quarter (Barri Gotic), which is the real old city, is accessible by wheelchair, and there are no high places where you need to climb up. However, during the summer it is usually extremely crowded, and it is full of narrow streets which can cause difficulties for wheelchair users. Thus, it is highly recommended for people with disabilities to visit this area in the morning, before it becomes overcrowded.
The biggest part of Ciutat Vella and the metro stations situated in the area are wheelchair-friendly. Of course, there are a few exceptions (for example station Jaume I, L4), therefore It is recommended to arrive to this area by bus which is fully accessible by wheelchair (for additional information – Public Transportation in Barcelona).