How the heck am I supposed to beat lunatic chain challenge?

You don't. You wait. You build an army. You conquer.
I haven't completed all the chain challenges myself and shamefully haven't put the effort into squad assault...
You'll get there.

A mix of repositions, a self healing character, a dancer and a reinhardt or brave lyn. If you have those I think you can do all lunatic challenges as of now.

I clear them all with one team, a bunch of fliers all with reposition and renewal 3. (Cleared all the challenges on lunatic with that easily no deaths)

You need to build a really well-balanced team/teams with good movement and sustainability. I can't really give you an in-depth guide to Chain Challenge, but I can tell you good units and skills to use.

1. Movement-based skills like Reposition, Draw Back. Positioning skills that let teammates get in, take out enemies, and get out of the danger zone are key to survivability. They can also be used to extend your units movement range by having teammates reposition or drag them along the map.

2. Healers or Pseudo-healers. Healers are nice since they can sustain your team to survive more fights. Self-healing units with Renewal and skills that swap HP like Ardent Sacrifice and Reciprocal Aid can give you more mileage than a traditional healer since they can also be used to take out enemies. A popular build for this is the Falchion + Renewal 3 combat medic build with characters like Chrom, Lucina, Masked Marth, etc.

3. Dancers. Dancers are fantastic support units that let your team sweep enemies or even escape danger after taking out threats.

Personally I have a cavalry emblem team that usually gets me at least most of the way through. Xander with QR, cavalry buffs, and reposition added to his base set, Oscar with his base set+armored blow and Moonbow, Cecilia with Moonbow, reposition, TA1, and Cavalry buffs, and Priscilla on her base set with cavalry buffs and live to serve. I usually lean towards defensive buffs since there is no time limit and losing one member can make it all fall apart.

People do it with only 4 star units.

The trick is to just use one really powerful team. I usually use one team for all of the battles and maybe finish off the last one with a crappy unorthodox team. You definitely need a sweeper (quabridelia, quadelia, brave lyn, or a reinhardt) with a dancer. You do not need a self healing character, instead put breath of life and maybe even a breath of life seal on your sweeper. You will be solely using the sweeper to clear the whole map if possible. A powerful dc unit is necessary to bait nuisances, reposition the sweeper out of harm, and tank on enemy phase. Heal the dc unit with breath of life. That's all you really need. The fourth unit can be a mage baiter/repositioner.

1) A combat medic is a huge help for CC so you don't have to have a unit not able to do anything when not healing. If you have Masked Marth from the last TT, build her out with Renewal 3 and Ardent Sacrifice and she can heal for 10 HP a turn while still being able to kill reds and greens for you.

2) Triangle Adept helps a lot so your physical tanks don't need to be healed as much.

3) A ranged unit who gets into a full glass canon mode is key to doing most of the killing. Reinhardt, Desperation Brave Bow users, Nino: whatever happens to be best on your team. You have to havea Nino you can build up at this point.

In my case, my team is a Horse emblem with Luke, Camus and Reinhardt, with hone and spur calvary. My 4th hero is a healer, in my case Lissa, with rehabilitate and wings of mercy.

Bridelia, fury+vantage camus, qr3 ike, rehabilitate healer.
Ike and camus have reposition. Healer has live to serve

Bridelia, fury+vantage camus, qr3 ike, rehabilitate healer.
Ike and camus have reposition. Healer has live to serve

Bridelia, fury+vantage camus, qr3 ike, rehabilitate healer.
Ike and camus have reposition. Healer has live to serve

horsey emblem, heavy on player phase one-shotting and positioning supports, to avoid taking hits if possible. I.e. Rein, Olwen, Cecilia, Blyn, Roderick, maybe you can run a Broy or Eldigan/Xander to have a bit more mix in there. Otherwise, forting up and healing up similar to TT runs.

Like everyone else said you really need 1-2 powerful highly synergized teams to pull it off... and patience. Horse emblem teams are probably the easiest to start out with since they get massive buffs, but I like infantry teams better.

I've found that a blade tome + self-healing tank (aether) + dancer (healing dancer is best) + another unit of a complimenting color(all with hone/fortify/spurs) does the job nicely. With the exception of 4 maps (stupid turn limit maps with celica and boey on defense tiles... hnggggg) I've been able to beat them all after a couple tries. Since the team is centered around my blade tome, if he dies it screws my whole run. If I can keep him alive, then he can essentially wipe out 2 units per turn, then my tank can reposition him out of the way, take a hit (maybe get a kill) and then have the blade tome finish off the rest.

Honestly, that's all it is. You have to build one team that is ABSOLUTELY meant to work together; like you take out one part and it no longer works. Can't just throw 4 strong units together here and hope for the best, unfortunately.