From what I remember, the two sacraments that do not need an ordained cleric in extraordinary circumstances are baptism and marriage. Am I correct?
Canon law makes provisions for this, but it would be very rare. Catholics must contract marriage beore the local ordinary or pastor. Or, a priest or deacon delegated to assist. Or, a lay person delegated by the bishop to assist. They may contract in front of witnesses only when there is danger of death or a situation in which no representative of the Church can be called to witness and that situation will persist for more than one month.
Can. 1108 §1. Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses according to the rules expressed in the following canons and without prejudice to the exceptions mentioned in cann. ⇒ 144, ⇒ 1112, §1, ⇒ 1116, and ⇒ 1127, §§1-2.
§2. The person who assists at a marriage is understood to be only that person who is present, asks for the manifestation of the consent of the contracting parties, and receives it in the name of the Church.
Can. 1109 Unless the local ordinary and pastor have been excommunicated, interdicted, or suspended from office or declared such through a sentence or decree, by virtue of their office and within the confines of their territory they assist validly at the marriages not only of their subjects but also of those who are not their subjects provided that one of them is of the Latin rite.
Can. 1110 By virtue of office, a personal ordinary and a personal pastor assist validly only at marriages where at least one of the parties is a subject within the confines of their jurisdiction.
Can. 1111 §1. As long as they hold office validly, the local ordinary and the pastor can delegate to priests and deacons the faculty, even a general one, of assisting at marriages within the limits of their territory.
§2. To be valid, the delegation of the faculty to assist at marriages must be given to specific persons expressly.
If it concerns special delegation, it must be given for a specific marriage; if it concerns general delegation, it must be given in writing.
Can. 1112 §1. Where there is a lack of priests and deacons, the diocesan bishop can delegate lay persons to assist at marriages, with the previous favorable vote of the conference of bishops and after he has obtained the permission of the Holy See.
§2. A suitable lay person is to be selected, who is capable of giving instruction to those preparing to be married and able to perform the matrimonial liturgy properly.
Can. 1113 Before special delegation is granted, all those things which the law has established to prove free status are to be fulfilled.
Can. 1114 The person assisting at marriage acts illicitly unless the person has made certain of the free status of the contracting parties according to the norm of law and, if possible, of the permission of the pastor whenever the person assists in virtue of general delegation.
Can. 1115 Marriages are to be celebrated in a parish where either of the contracting parties has a domicile, quasidomicile, or month long residence or, if it concerns transients, in the parish where they actually reside. With the permission of the proper ordinary or proper pastor, marriages can be celebrated elsewhere.
**Can. 1116 §1. If a person competent to assist according to the norm of law cannot be present or approached without grave inconvenience, those who intend to enter into a true marriage can contract it validly and licitly before witnesses only:
1/ in danger of death;
2/ outside the danger of death provided that it is prudently foreseen that the situation will continue for a month.
§2. In either case, if some other priest or deacon who can be present is available, he must be called and be present at the celebration of the marriage together with the witnesses, without prejudice to the validity of the marriage before witnesses only.**
Can. 1117 The form established above must be observed if at least one of the parties contracting marriage was baptized in the Catholic Church or received into it and has not defected from it by a formal act, without prejudice to the prescripts of ⇒ can. 1127, §2.
Also, can I begin a discussion about why cohabitating couples have not sacramentally married themselves?
Why do you think they have? See canons above, plus other canons pertaning to valid exchange of consent. Does cohabiting meet these requirements? No.
Finally, have they excommunicated themselves?
Cohabitation is not listed as an offense that incurs Latæ sententiæ excommunication. So, no.
They are, however, in a state of grave sin against the Sixth Commandment. They have separated themselves from the Eucharist. They should cease cohabiting and go to Confession as soon as possible.