No dimension is tangible.
It is one of the most frustrating aspects of science, the fact that otherwise eminent academics have allowed the word dimension to be so badly misused. (And, yes, Einstein must take some of the blame – but mostly I blame those who came after him)
Dimension means measurement.
A dimension is a measurement framework that we overlay an aspect of reality to calibrate and reference that aspect of reality.
Example – Temperature is the dimension of heat. Temperature is how we calibrate and index heat. Temperature is abstract, heat is real.
Space and Time are both words with multiple meanings. Both have a context as dimensions.
In the context of dimensions, Space is the dimension of (relative) static position, hence xyz-axis. The dimension of Space is how we measure relative static position, using standard units. It is static relative position because, by definition, it excludes Time.
The dimension of Time is how we measure, calibrate, index change. For instance, changing position (motion) is calibrated in, say, miles per hour. The per hour bit is time calibrating the rate of change.
So merge Time, the dimension of change, and Space, the dimension of relative position, and we get Spacetime, the dimension of relative changing position (motion) (in fact relative constant motion, as dimensions are reference-frame specific).
So, in answer to you question, the fourth dimension, like all dimensions, is NOT tangible.