Are science and spirituality mutually exclusive?

No.
Science and spirituality are indeed mutually inclusive.
In this context, we don’t talk about science facts, because science facts can be deceiving at sometimes due to some experimental errors, and slanted observations, but in here we should rather look at science itself.Spirituality has often been associated with religion, and therefore the connection between science and spirituality has been obscured to many. Spirituality is about self realization, whereby, you learn about yourself (includes your inner self), and there are various ways in pertaining it, but all of the ways will strictly require us to know and get acquainted to science. For e.g.; meditation. Meditation is said to be the most ideal pathway into self realization and unraveling the true potential of oneself. But meditation is not a one step process, especially for those who intend to seek for the highest level of meditation. One should have dominance on his or her respiration, temperature, anxiety (by the regulation of hormones) and etc. All these are biological processes, a sub discipline of science. All these fascinating stuff, you will only be able to do it if you unlock your subconscious mind.
Therefore, science and spirituality co-exist together in a harmonious relationship, whereby the latter having an effect on another and vice versa.