Do any Christian groups sell all their possessions as it seems the Early Church did?

The answer to this question can really, literally, only be answered with "it depends on who you ask". Before throwing a downvote at me, read on though:

We see in Acts 2:45 one of the reasons they gave to the poor - they had need - which reflects Christ teaching throughout the Gospels (as well as many OT passages) to fulfill the needs of those in need:

And sold their possessions and goods, and parted them to all men, as every man had need. - Acts 2:45

“If a man shuts his ears to the cry of the poor, he too will cry out and not be answered.” - Proverbs 21:13

In 1 John 3:17 we see the same reflected

“If anyone has material possessions and sees his brother in need but has no pity on him, how can the love of God be in him?”

And in Job

“I rescued the poor who cried for help, and the fatherless who had none to assist them” (Job 29:12)

So we see it one of two ways, really:

either a) no, (to my knowledge) no denomination actively sells their possessions. I had asked a pastor this before, and in his opinion this was because of the incident with Jesus and the scourge in the temple in John 2. Since selling implies the exchange of money (at least, again, in his opinion).

or b) if you consider regular charity, giving to those in need, and fundraising to be fulfilling of the verses quoted in your question, then there are many that actively do this. Sometimes these are personal possessions, sometimes they are possessions given to them by the community.

I agree with @Flimzy's excellent reply, but I see he didnt answer your question completely, and since I can't comment (not enough reputation) I will add this reply.

Regarding living arrangements. They used large apartment buildings where families had their own living room and bedrooms, and then shared kitchens (sometimes private) and public restrooms (male and female separate). They did NOT have everything in common, for example their wives!

When I visited one group in the 1980's they were struggling in how to include an African Bruderhof group, as their standard of living would be much less then they were accustomed to, and they didn't know how they could live at a lower standard of living, or how to justify having a higher standard of living than their "brothers." A very difficult question, I don't know how they resolved it.

They made group decisions by consensus of the whole community of perhaps 600 people! They had a difficult time wrestling with if they should take and eat the free ice cream pies that a local factory produced that were flawed, but healthy and scrumptious, but not simple.

Yes, the Bruderhof Communities do that, although with some practical limits such as having personal clothing and toiletries. Their beliefs are outlined in this document that states:

We are a fellowship of brothers and sisters, both single and married, who are called by Christ to follow him together in a common life in the spirit of the first church in Jerusalem. We desire to remain true to this calling all our lives. For its sake we gladly renounce all private property, personal claims, and worldly attachments and honors. Our vocation is a life of service to God and humankind, freely giving our whole working strength and all that we have and are.

Bruderhof members do not live as monks and nuns but are a congregation of families, children, and single people who live on their community locations worldwide. While some people confuse them with more reclusive groups such as the Amish, they run a publishing house and welcome visitors as well.

If you have seen the movie The Mission, then you may have noticed a communal society lived by the natives in the various missions led by Jesuit priests. Historical records corroborate that indeed they lived a Christian-based, communal life. From Jesuit Reductions:

Smaller missions had two priests, whereas larger missions had more. Populations varied from 2,000 to 7,000. In the morning, children's hymns were followed by Mass and breakfast, after which the workers went to their tasks. "The Jesuits marshaled their neophytes to the sound of music, and in procession to the fields, with a saint borne high aloft, the community each day at sunrise took its way. Along the way at stated intervals were shrines of saints where they prayed, and sang hymns between shrines. As the procession advanced it became gradually smaller as groups of Indians dropped off to work the various fields and finally the priest and acolyte with the musicians returned alone" (Graham, 178–9). At noon each group assembled for the Angelus, after which came dinner and a siesta; work was then resumed until evening. After supper came the rosary and sleep. On rainy days they worked indoors. Frequent festivals with sham battles, fireworks, concerts, and dances enlivened the community.

Aside from the main farm, each man typically had his own garden, pursuing agriculture, stock raising, and the cultivation of maté. Jesuits introduced many European trades and arts to their communities. Cotton weavers, tanneries, carpenters, tailors, hat makers, coopers, boat builders, silversmiths, musicians and makers of musical instruments, painters, and turners could sometimes be found. They also had printers, and manuscripts were also produced by hand copying.(Graham)

The goods that were produced at the missions, including cattle, were sold in Buenos Aires and other markets under the supervision of the priests. The proceeds earned were divided among a common fund, the workers, and dependents.

Note also the combination of communal and individual labor and the variety of trades pursued.