It may seem ‘old school’ but YES business cards are absolutely still relevant in 2020.
When you go to a networking event you're there selling yourself, not the massive company with hundreds of people behind you, or your comfortable small office with 1 to 2 members of staff, you’re selling you to the person in front of you.
I don’t know about you but I get inundated with sales email after sales email with people constantly giving me their business information - this is what we do, this is how we can help you etc. Most of those emails get sent straight to trash. So if I meet you at the networking events and you tell me you can email me over your details, chances are the second I leave the room not only have I forgotten your name (and not in a rude way but just because I've met so many people that day) chances are by the time your email come through it's lost in the abundance of sales email I get which are uninvited and end up in the trash.
However, you hand me your business card and it's a whole different story. That business card represent you as a brand. It immediately tells me the the kind of person you are. I can tell the quality of your company, how much you care about your business, The time and effort you put into things. If you have made your own business card on your laptop and printed it from a Word document I know you don't invest in yourself. But if you hand me a quality well-designed business card that immediately tells me that you are a person who take themselves seriously.
I can honestly say if I take your business card it's because I genuinely want to get in contact with you after that events.
So yes, in my opinion, Business Cards are very much still relevant in 2020.