Approaching a professor for a postdoc, after not taking his previous postdoc offer in favor of an industry job I had a postdoc offer from a great research group at a US university towards the end of my PhD in April 2018. Around that time, a family member became severely ill. The nature of the illness was such that the doctors told me that they had to wait a few months to get a clear diagnosis and know whether it was going to be a long term condition or a one-off event. This put me in a dilemma - should I find a full time job that would let me relocate close to my family overseas if need be or should I stay in academia and hope it was just a one-off event? I decided to buy some time and wait a few months and never got back to the professor about the offer - I was very embarrassed to mention the illness as it's a highly stigmatized one. Morever, the professor wanted me to join immediately after my PhD and did not seem to be very flexible about my joining date. He never got back in touch after not hearing from me - I suppose he retracted the offer a few weeks after he made it. 

I ultimately ended up delaying my PhD defense by a few months and found a corporate software job after that and started in February this year. My mental health has been slowly declining due to the fast-paced, extremely profit-driven nature of the corporate work environment and I am actually thinking of quitting and going back to a postdoc (my parents have decided to look after the sick family member so the illness is not as immediate a concern as it previously was).  I apologized to the professor in March about my non-response regarding his postdoc offer and told him about the illness and that I have moved on to the industry and he was very understanding and wished me all the best. 

Here is my question: I know he is still looking for postdocs. How do I even begin the conversation with him on joining his group? I am also a bit concerned that the 1 year gap on my academic CV will be looked down upon negatively in future job searches. 

If it is important to this question at all, I am 27 yrs old.
Start the conversation by being honest: describe what had happened, and assure them that it's only because of personal reasons that you could not accept their offer, not just the industry offer. You did the right thing by emailing them in March to keep them in the loop. However, I would emphasize that the reason that you didn't email them sooner was because of the personal stuff.

The same reasoning goes for your 1 year gap: if you do amazing work in your postdoc all will be forgiven. You need to convince the advisor that they can trust you to do well despite the initial setback (which was not something you could control), and the only way I can think of doing this is by telling them the truth. 

I had a collaborator (not a postdoc) who left academia for personal reasons and returned to work with me (their story is somewhat similar to what you're describing, though the personal circumstances are different). I was very happy with them and was understanding regarding their situation. I'd like to think that I'm not an exception: most reasonable people understand that their students/postdocs have a life outside academia, and this particular advisor sounds like they should be as well. 

Your age is not a major factor in this equation. I'd say that 27 is relatively young (I started my PhD at the ripe old age of 28).