Are there any examples of people who successfully did a startup AND worked in academia? Or is it better to just choose one?
There is a huge literature on the topic of academic entrepreneurs (and entrepreneurial academics). A recent review of the literature can be found here. Similarly, there are many books on the topic (e.g. here and here). Plus millions of blog posts (e.g. here).

I think it is hard to generalise. For example, this paper highlights the multitude of dimensions involved in the extent and success of entrepreneuring, some outside the control of the individual. This paper distinguished between three groups. First, the ‘entrepreneurial academics’:


  These are individuals who planned to stay in a university but also spent time working in spin‐out companies. They had experienced a number of operational difficulties in trying to develop this new career type and many felt that although the government and universities said they want them to work like this, they had been slow to adjust their own policies and procedures to take account of this change.


A second group, consisting of:


  individuals who were in the process of developing a new kind of career as technology‐transfer professionals. Notably they were younger than those in the first group, and more junior within their departments. The majority wished to stay working in a university context but did not see themselves staying at the laboratory bench. 


Finally, the third group:


  made up of younger people who have either recently completed their PhD or have been employed on postdoctoral research contracts. Most seemed unsure where their future direction lay and were using [a research program] to develop the knowledge, skills and resources that they saw as vital to the development of a successful career. ... The scientists in the third group were well aware of the traditional scientific career path and its status but saw it as largely unavailable; a relic of the past that was now only offered to the lucky few. Their aim was to find, or develop, a career path that would allow them to use their science, provide opportunities for growth, and offer a degree of security unavailable to the large numbers of postdoctoral scientists who find themselves in a seemingly endless spiral of short‐term contracts.


Relatedly, this paper offers and tests a model of career path, concluding that:


  entrepreneurial self-efficacy, type of research, perceived role models, number of years spent at an academic institution, and patents are significantly related to the formation of academic-entrepreneurial intentions, regardless of cultural context.


A similar approach is taking by this paper.

The bottom line is that success as entrepreneurs depends on many conditions, some outside the control of the academic. There are different models in which academics engage in these activities, some involving moving away from academia. All in all, beyond anecdotal evidence, of course there are successful entrepreneurial academics. Whether you can be one or not is something you must confront with the evidence above and your own insights.