If you mean non-Christians, the answer is clearly “no”.

But, assuming you mean other Christians, the short answer is “yes”. The slightly longer, but more accurate answer, is “yes, IF…”

It seems pretty basic, but you might be surprised how often this question still comes up, and worse, how often the people offering answers get it wrong, at least in part.

The question asks, “is it possible”, not “is it normally the case”. The answer to the former is “yes”, and to the latter “no, but there are exceptions”.

One thing that is absolutely clear: To say “non-Catholics cannot receive sacraments in the Catholic Church” is plainly wrong according to the law, and potentially sinful. 

First let us remember the ideal: That all Christians should be part of one and the same communion, that One Church willed by Christ in a real, visible, tangible way. In such a case, naturally all Christians could receive communion together.

However, because of our brokenness, because of our human failing and sin, because of fault that lay with Catholics, Orthodox, and Protestants alike, the Church is wounded. Its unity maybe substantially present, but it is defective, and needs repair. As long as this abnormal state of division persists, we cannot freely share Eucharistic communion, which is a sign of ecclesiastical communion.

Above all, this not-sharing is meant to provoke a painful longing that prompts action for unity.

In 1983, the revised Code of Canon Law for the Latin Church was published, nearly twenty years after the Council that mandated it. It outlined several conditions for the ministry of the sacraments in Canon 844. Allow me to summarize:

§844.1  You should normally receive sacraments only from ministers in the same communion. But there are exceptions built into this norm. (That is, as part of the law itself; not even considering exceptions of pastoral prudential judgement or oikonomia)

§844.2 Catholics can always receive some sacraments from some non-Catholic Churches.

§844.3 Some Non-Catholic Christians are always welcome to participate in some sacraments from the Catholic Church.

§844.4 Non-Catholic Christians who do not fall into the category of always being welcome to participate in these sacraments, can be allowed sometimes, in certain situations. 

§844.5 Local or national norms must be made in consultation with ecumenical counterparts in any affected church or communion. 

There are certain considerations that apply universally to this question, whether for Catholics or non-Catholics approaching the sacraments:

• Only baptized Christians can approach the sacraments
• Proper disposition is always expected
• It is a free and spontaneous act, motivated out of genuine need or desire for the sacrament (as opposed to an act of protest or a ‘shotgun’ sacrament, for example)
• “Indifferentism” and “Triumphalism” must both be avoided
  • Indifferentism is the sin of accepting our divisions as normal, and that sacramental sharing between broken communions is normal or normative.
  • Triumphalism is the sin of thinking that ‘we’ are better than the other because we ‘own’ the truth, or the Real Presence, or suchlike. Making mockery of other churches’ liturgies or sacraments is an example.

§844.3 and .4 are the most relevant to the original question:
Can non-Catholic Christians receive communion at a Catholic church?

Paragraph three tells us that, given the universal conditions above, the members of the following churches and communions are always welcome at Catholic sacraments:  Eastern Orthodox, Oriental Orthodox, Assyrian Church of the East, Old Catholics; anyone else the pope determines to be “in a similar situation”.

Paragraph four tells us that anyone not in the “always welcome” category can receive Reconciliation, Eucharist, or Anointing under the following circumstances:

-They share the Catholic faith in these sacraments  (e.g., believe in the Real Presence of Christ in the Eucharist), AND
-Their own minister or community is inaccessible, AND EITHER
-They are in danger of death, OR
-They have a grave or pressing need (What these are to be determined by the bishop or bishops’ conference. Those that have done so have named, e.g., weddings, funerals, a child’s first communion, mixed marriages, spiritual crises, et al.).

And that remains the law in force. However, just a dozen years after the promulgation of this code, the same pope who authorized it, St. John Paul II, modified these requirements – and this is frequently overlooked.

In his 1995 encyclical Ut Unum Sint, the Polish pontiff said,

…it is a source of joy to note that Catholic ministers are able, in certain particular cases, to administer the Sacraments of the Eucharist, Penance and Anointing of the Sick to Christians who are not in full communion with the Catholic Church but who

1) greatly desire to receive these sacraments,

2) freely request them, and

3) manifest the faith which the Catholic Church professes with regard to these sacraments. (UUS, §46)

First, it is a “source of joy”. Not grudging admission or acceptance. Not mere tolerance nor haughty triumphalism. It is a a joy that we can share the sacraments, some of the time, despite our divisions.

It is just a pity he did not actually modify the Code to match his encyclical, which would have removed all doubt – but that was a common problem of his pontificate.

Not only is saying “Non-Catholics cannot receive sacraments in the Catholic Church” plainly wrong according to the law, it is morally wrong to take any joy in denying fellow Christians this opportunity or from the state of brokenness in which we find ourselves.

Other developments between 1983 and 1995:

• “danger of death or grave and pressing need” is simplified to “great desire”
• “spontaneously request” is clarified as “freely” requesting, so no one is tempted to hinder someone who has been thinking about this for a while
• the accessibility of their own minister is irrelevant

That was twenty years ago. It is only to be expected that the fruits of ecumenical dialogue have resulted in even further development. Pope Francis, in his visits to the Lutheran and Anglican congregations in Rome, and in his Jubilee recognition of Lefebvrite confessors, has indicated as much.

Perhaps it is time for the apostolic see to recognize “in a similar situation” to the Eastern churches, some of the churches and ecclesial communities of the West, particularly the Anglicans, some Lutherans, and the SSPX. At least, we can acknowledge the growth in agreement about sacraments, especially Orders and Eucharist, to be sufficient to allow more common sharing along the lines of John Paul II’s vision of twenty years ago, or more.

[TL;DR]:

So, can non-Catholics receive communion at a Catholic Eucharist? Yes, they can – if they are baptized, properly disposed, recognize that this division of Christians is not normal, greatly desire them, freely request them, and share a “catholic” belief about them.

Which, if you think about it, would likely be the case if someone is intentionally approaching the sacrament in a Catholic Church anyway, no?