Yes and no.
To speak plainly. The CDC order is for basically every renter of a residential property. It prohibits evictions from September 4, 2020, until the end of the year, December 31, 2020.
This means that tenants can have protection under federal guidelines from being evicted during this time period. It is worth mentioning, however, that, just like in the Massachusetts law, the federal law does not allow tenants to forego paying rent. In fact, in its FAQ worksheet, the CDC states:
“Covered people still owe rent to their landlords. The Order halts residential evictions only temporarily. Covered persons still must fulfill their obligation to pay rent and follow all the other terms of their lease and rules of the place where they live.”
Thus, the goal is simply not to have people homeless or having to resort to overcrowded living situations or to situations that would put their health or that of their neighbor’s or of the community-at-large at risk.
There is one notable difference between the Mass moratorium and the federal one. The Mass is “state-centered.” It is automatic, requiring nothing of the tenant in order to enjoy its protections. The CDC moratorium requires an affirmative act on the part of the tenant to enjoy the protections. Namely, the tenant must provide to the landlord a “declaration under perjury” that s/he has done one of a list of things.
For a more nerdy analysis, consider that the moratorium, whether from the Governor or from the CDC, a federal agency, is something of an economic taking; in that, debtors owed money are not being able to carry out remedies (like eviction) contracted for. This would qualify as an infringement on their constitutional right to enter into contracts.
The law has seemed quite clear on this point. Since the days of Lochner, a 1905 case protecting the freedom of contract under the due process clauses of the 5th and 14th Amendments, the courts have ruled in favor of just about every law restricting freedom of contract due to governmental interest. This case ushered in a whole era of cases lasting up to 1937, which period is generally know as the Lochner Era. This was a time that the Court decided that government should get out of the way of people entering into contracts or agreements. During this period, hundreds of cases resulting in findings that the disputed laws were unconstitutional infringements of the freedom of contract.
Nowadays, this jurisprudential philosophy has been turned on its head. Now, any laws regulating business will be found constitutional under the due process clause as long as they are rationally related to serve a legitimate government purpose. This is a very easy test for the government to pass, as it does not need to show the law drafted is necessary to achieve a compelling interest. (In this case, the government interest is undoubtedly its interest in protecting the health of the citizens). It doesn’t even have to show that the law is important or substantially related to an important government purpose (the test for when a State or local authorities is alleged to have violated a person’s liberty to contract. In other words, is the law a reasonable way of accomplishing the goal? Put differently, is the moratorium (and the resultant inability of landlords to evict as per a contractually agreed-upon remedy) a reasonable way of accomplishing the protection of its citizens? The answer so far, at least since 1937, has been, undoubtedly, yes.
Of course, the CDC has express statutory authority as well:
Whenever the Director of the Centers for Disease Control and Prevention determines that the measures taken by health authorities of any State or possession (including political subdivisions thereof) are insufficient to prevent the spread of any of the communicable diseases from such State or possession to any other State or possession, he/she may take such measures to prevent such spread of the diseases as he/she deems reasonably necessary, including inspection, fumigation, disinfection, sanitation, pest extermination, and destruction of animals or articles believed to be sources of infection.