What style of hot rod has a short wheelbase and open engine?

The hot-rod style with a short wheelbase and an open engine is a T-bucket.

A T-bucket is a heavily modified hot rod based on the two-seat roadster body of a Ford Model T, especially the 1915–1927 models. The body is typically fenderless or minimally fendered, with the front wheels exposed and the engine prominently visible ahead of the cockpit. Its compact proportions give the style the short, dramatic stance described in the question.

The name comes from the Model T’s small, bucket-shaped roadster body. Although Model Ts had been modified since the 1920s, the specifically recognizable T-bucket style emerged in the 1950s. Norm Grabowski’s “Kookie Kar,” built from a Model T-based car and popularized by the television series 77 Sunset Strip, helped make the design widely known.

T-buckets are commonly fitted with large V8 engines, wide rear tires, and narrow front wheels. That combination distinguishes them from gassers, which are usually altered production cars with a raised front end, and rat rods, which are defined more by deliberately rough or unfinished appearance than by a particular Model T body style.