Veneer. At least the top anyway. You will notice that it is symmetric along two axes. Which means it has a four way book match which is something that is really only possible with veneer.
handles can be replaced- not always a good indictor of age. My 2010's bedroom set has original 1930s hardware from other dressers. in fact all my dresser types-inc. benches have old hardware. my guess-30's-50s. quality manf. Few nice pieces are solid- veneer has been around for a couple centuries.